# Fast and Accurate Computation of Gauss-Legendre and Gauss-Jacobi Quadrature Nodes and Weights

@article{Hale2013FastAA,
title={Fast and Accurate Computation of Gauss-Legendre and Gauss-Jacobi Quadrature Nodes and Weights},
author={Nicholas Hale and Alex Townsend},
journal={SIAM J. Sci. Comput.},
year={2013},
volume={35}
}
• Published 6 March 2013
• Computer Science
• SIAM J. Sci. Comput.
An efficient algorithm for the accurate computation of Gauss--Legendre and Gauss--Jacobi quadrature nodes and weights is presented. The algorithm is based on Newton's root-finding method with initial guesses and function evaluations computed via asymptotic formulae. The $n$-point quadrature rule is computed in $\mathcal{O}(n)$ operations to an accuracy of essentially double precision for any $n\geq 100$.
142 Citations

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## References

SHOWING 1-10 OF 69 REFERENCES

### O(1) Computation of Legendre Polynomials and Gauss-Legendre Nodes and Weights for Parallel Computing

• Computer Science
SIAM J. Sci. Comput.
• 2012
A self-contained set of algorithms is proposed for the fast evaluation of Legendre polynomials of arbitrary degree and argument and immediately yields an $\mathcal{O}(1)$ algorithm for computing an arbitrary Gauss--Legendre quadrature node.

### A Fast Algorithm for the Evaluation of Legendre Expansions

• Computer Science
SIAM J. Sci. Comput.
• 1991
An algorithm is presented for the rapid calculation of the values and coefficients of finite Legendre series and admits far-reaching generalizations and is currently being applied to several other problems.

### Accurate Computation of Weights in Classical Gauss-Christoffel Quadrature Rules

A simple but efficient and general method for improving the accuracy of the computation of the quadrature weights is proposed, which ensures a high level of accuracy for these weights even though the nodes may carry a significant large error.

### A Survey of Gauss-Christoffel Quadrature Formulae

We present a historical survey of Gauss-Christoffel quadrature formulae, beginning with Gauss’ discovery of his well-known method of approximate integration and the early contributions of Jacobi and

### Algorithm 726: ORTHPOL–a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules

A collection of subroutines and examples of their uses, as well as the underlying numerical methods, are described for generating orthogonal polynomials relative to arbitrary weight functions. The

### On the convergence rates of Legendre approximation

• Mathematics
Math. Comput.
• 2012
The problem of the rate of convergence of Legendre approximation is considered, and explicit barycentric weights, in terms of Gauss-Legendre points and corresponding quadrature weights, are presented that allow a fast evaluation of the Legendre interpolation formula.

### NEWTON’S METHOD IN FLOATING POINT ARITHMETIC AND ITERATIVE REFINEMENT OF GENERALIZED EIGENVALUE PROBLEMS∗

It is shown that iterative refinement by Newton’s method can be used to improve the forward and backward errors of computed eigenpairs and bound the limiting accuracy and the smallest norm of the residual.